Symmetries of Monocoronal Tilings

Symmetries of monocoronal tilings

Dirk Frettl¨oh

Technische Fakult¨at Universit¨atBielefeld

Geometry and Symmetry 29 June - 3 July 2015, Veszpr´em

Dedicated to K´arolyBezdek and Egon Schulte on the occasion of their 60th birthdays

Dirk Frettl¨oh Symmetries of monocoronal tilings Dirk Frettl¨oh Symmetries of monocoronal tilings 1. Monohedral and isohedral tilings 2. Monogonal and isogonal tilings 3. Monocoronal (and isocoronal) tilings

Joint work with Alexey Garber.

Dirk Frettl¨oh Symmetries of monocoronal tilings 2 Tiling (=tessellation) covering of R which is also a packing. Pieces (tiles): nice compact sets (squares, triangles...).

Dirk Frettl¨oh Symmetries of monocoronal tilings 2 Here: tilings of R . 2 Def.: The symmetry group of a tiling T of R :

2 Sym(T ) = {ϕ isometry in R | ϕ(T ) = T}

A central question: Which shapes do tile?

2 I Euclidean plane R d I Euclidean space R (d ≥ 2) 2 I hyperbolic space H I ﬁnite regions like 2, 4, ...

Dirk Frettl¨oh Symmetries of monocoronal tilings A central question: Which shapes do tile?

2 I Euclidean plane R d I Euclidean space R (d ≥ 2) 2 I hyperbolic space H I ﬁnite regions like 2, 4, ... 2 Here: tilings of R . 2 Def.: The symmetry group of a tiling T of R :

2 Sym(T ) = {ϕ isometry in R | ϕ(T ) = T}

Dirk Frettl¨oh Symmetries of monocoronal tilings Monohedral and isohedral tilings

Dirk Frettl¨oh Symmetries of monocoronal tilings Image: isohedral (hence monohedral) tiling.

A tiling is called monohedral if all tiles are congruent. A tiling T is called isohedral if its symmetry group acts transitively on the tiles.

Dirk Frettl¨oh Symmetries of monocoronal tilings A tiling is called monohedral if all tiles are congruent. A tiling T is called isohedral if its symmetry group acts transitively on the tiles.

Image: isohedral (hence monohedral) tiling.

Dirk Frettl¨oh Symmetries of monocoronal tilings Image: another isohedral and monohedral tiling.

Dirk Frettl¨oh Symmetries of monocoronal tilings Some isohedral and non-isohedral tilings:

...?

Dirk Frettl¨oh Symmetries of monocoronal tilings The three species hexagons allowing monohedral tilings: Type 1 Type 2 Type 3 akd A + B + D = 2π A = C = E = 2π/3 a = d a = d, c = e a = b, c = d, e = f

f a E e d A e D c B b C d A c a

2 The list of all isohedral tilings of R by convex polygons is known (Reinhardt 1918, see also Gr¨unbaum and Shephard 1987)

Dirk Frettl¨oh Symmetries of monocoronal tilings 2 The list of all isohedral tilings of R by convex polygons is known (Reinhardt 1918, see also Gr¨unbaum and Shephard 1987)

2 The list of all convex polygons allowing monohedral tilings of R is maybe incomplete. 2 I All triangles can tile R by congruent copies. 2 I All quadrangles (convex or non-convex) can tile R . 2 I There are three kinds of convex hexagons that can tile R . The three species hexagons allowing monohedral tilings: Type 1 Type 2 Type 3 akd A + B + D = 2π A = C = E = 2π/3 a = d a = d, c = e a = b, c = d, e = f

f a E e d A e D c B b C d A c a

Dirk Frettl¨oh Symmetries of monocoronal tilings 2 A convex n-gon cannot tile R by congruent copies for n ≥ 7. Only open case: pentagons. Below a list of 14 species of convex 2 pentagons that can tile R by congruent copies. It is unknown whether this list is complete.

...?

Dirk Frettl¨oh Symmetries of monocoronal tilings Monogonal and isogonal tilings

Dirk Frettl¨oh Symmetries of monocoronal tilings The classiﬁcation of all isogonal tilings is known, see

Gr¨unbaum& Shephard: Tilings and Patterns

A classiﬁcation of all monogonal tilings seems out of reach.

A tiling is called monogonal if all vertex stars (=vertex plus adjacent edges) are congruent.

A tiling is called isogonal if the symmetry group acts transitively on the vertex stars.

Dirk Frettl¨oh Symmetries of monocoronal tilings A tiling is called monogonal if all vertex stars (=vertex plus adjacent edges) are congruent.

A tiling is called isogonal if the symmetry group acts transitively on the vertex stars.

The classiﬁcation of all isogonal tilings is known, see

Gr¨unbaum& Shephard: Tilings and Patterns

A classiﬁcation of all monogonal tilings seems out of reach.

Dirk Frettl¨oh Symmetries of monocoronal tilings Monocoronal (and isocoronal) tilings

Dirk Frettl¨oh Symmetries of monocoronal tilings More restrictively, a tiling is called monocoronal wrt direct idometries if all vertex coronae in the tiling are congruent wrt direct isometries (i.e., no reﬂections allowed).

Monogonal, but not monocoronal.

The vertex corona of a vertex is this vertex together with its incident tiles. A tiling is called monocoronal if all vertex coronae in the tiling are congruent.

Example: All Archimedean tilings are monocoronal.

Dirk Frettl¨oh Symmetries of monocoronal tilings The vertex corona of a vertex is this vertex together with its incident tiles. A tiling is called monocoronal if all vertex coronae in the tiling are congruent.

Example: All Archimedean tilings are monocoronal.

More restrictively, a tiling is called monocoronal wrt direct idometries if all vertex coronae in the tiling are congruent wrt direct isometries (i.e., no reﬂections allowed).

Monogonal, but not monocoronal.

Dirk Frettlöh Symmetries of monocoronal tilings A classification of all monocoronal tilings was obtained in F-Garber 2015. It is known (see Grünbaum& Shephard) that any monocoronal tiling is of one of 11 combinatorial types:

3.3.3.4.4 3.3.3.3.3.3

3.3.3.3.6 3.3.4.3.4

Dirk Frettl¨oh Symmetries of monocoronal tilings 3.6.3.6 3.4.6.4

4.4.4.4

Dirk Frettl¨oh Symmetries of monocoronal tilings 3.12.12 4.6.12

4.8.8 6.6.6

Dirk Frettl¨oh Symmetries of monocoronal tilings In fact one may drop the requirements “convex” and “face-to-face”. This yields only 15 additional cases to consider. 2 Hence the results below hold for all monocoronal tilings of R .

···

Taking into account metrical properties one gets that:

There are 34 species of monocoronal face-to-face tilings by convex polygons (wrt diﬀerent vs equal edge length)

Dirk Frettl¨oh Symmetries of monocoronal tilings Taking into account metrical properties one gets that:

There are 34 species of monocoronal face-to-face tilings by convex polygons (wrt diﬀerent vs equal edge length)

In fact one may drop the requirements “convex” and “face-to-face”. This yields only 15 additional cases to consider. 2 Hence the results below hold for all monocoronal tilings of R .

···

Dirk Frettl¨oh Symmetries of monocoronal tilings In particular: If T is a monocoronal tiling wrt direct isometries, then

I T is crystallographic (2-periodic)

I T is vertex transitive (isocoronal)

I T has a center of rotational symmetry of order at least 2 (Here we use orbifold notation to denote the 17 wallpaper groups. E.g., ∗442 denotes the symmetry group of the regular square tiling; 442 denotes its rotation group)

Theorem (F-Garber 2015) Every tiling that is monocoronal wrt direct isometries (no reﬂections allowed) has one of the following 12 symmetry groups:

∗632, ∗442, ∗333, ∗2222, 632, 442, 333, 2222, 4∗2, 3∗3, 2∗22, 22∗.

Dirk Frettl¨oh Symmetries of monocoronal tilings Theorem (F-Garber 2015) Every tiling that is monocoronal wrt direct isometries (no reﬂections allowed) has one of the following 12 symmetry groups:

∗632, ∗442, ∗333, ∗2222, 632, 442, 333, 2222, 4∗2, 3∗3, 2∗22, 22∗.

In particular: If T is a monocoronal tiling wrt direct isometries, then

I T is crystallographic (2-periodic)

I T is vertex transitive (isocoronal)

Dirk Frettl¨oh Symmetries of monocoronal tilings ← A monocoronal tiling that is not 2-periodic. In the vertical direction one may stack layers like ··· LRLLRLLLLRLLLLLLLLRLL ··· (L: layer slanted to the left, R: layer slanted to the right)

If we allow reﬂected copies of vertex coronae the situation becomes more diverse. Theorem (F-Garber 2015) Every monocoronal tiling (reﬂections allowed) is either 1-periodic, or its symmetry is one out of 16 wallpaper groups: any except ∗×. If such a tiling is 1-periodic then its symmetry group is one of four frieze groups: ∞∞, ∞×, ∞∗, or 22∞.

In particular, every monocoronal tiling is crystallographic or 1-periodic.

Dirk Frettlöh Symmetries of monocoronal tilings If we allow reflected copies of vertex coronae the situation becomes more diverse. Theorem (F-Garber 2015) Every monocoronal tiling (reflections allowed) is either 1-periodic, or its symmetry is one out of 16 wallpaper groups: any except ∗×. If such a tiling is 1-periodic then its symmetry group is one of four frieze groups: ∞∞, ∞×, ∞∗, or 22∞.

In particular, every monocoronal tiling is crystallographic or 1-periodic. ← A monocoronal tiling that is not 2-periodic. In the vertical direction one may stack layers like ··· LRLLRLLLLRLLLLLLLLRLL ··· (L: layer slanted to the left, R: layer slanted to the right)

Dirk Frettlöh Symmetries of monocoronal tilings Theorem (F-Garber 2015) d For any d ≥ 3 there are non-periodic non face-to-face tilings of R that are monocoronal (reflections allowed). Theorem (F-Garber 2015) d For any d ≥ 4 there are non-periodic non face-to-face tilings of R that are monocoronal (reflections forbidden).

For dimensions d > 2:

Situation even more diverse. Depending on the exact assumptions (face-to-face or not, reﬂections allowed or not) the symmetry groups range from trivial to crystallographic.

Dirk Frettl¨oh Symmetries of monocoronal tilings For dimensions d > 2:

Situation even more diverse. Depending on the exact assumptions (face-to-face or not, reflections allowed or not) the symmetry groups range from trivial to crystallographic. Theorem (F-Garber 2015) d For any d ≥ 3 there are non-periodic non face-to-face tilings of R that are monocoronal (reflections allowed). Theorem (F-Garber 2015) d For any d ≥ 4 there are non-periodic non face-to-face tilings of R that are monocoronal (reflections forbidden).

Dirk Frettl¨oh Symmetries of monocoronal tilings Both results can be obtained by the following construction and its analogues in higher dimensions. Start with a 1-periodic plane tiling:

Dirk Frettl¨oh Symmetries of monocoronal tilings ...thicken it into a 3-dim layer:

...and stack these layers (some of them rotated) alternating with unit cube layers:

Dirk Frettl¨oh Symmetries of monocoronal tilings Regarding the third problem: there are non-periodic tricoronal tilings.

Open Problems

I Generalise Theorems 3 and 4 to face-to-face tilings. 2 I Consider the same problem for the hyperbolic plane H . I Consider the same problem for bicoronal tilings. In particular, is there a bicoronal nonperiodic tiling? Partial answers to the second problem can be found in F-Garber 2015, using B¨or¨oczkytilings. 2 In particular, there are monocoronal tilings in H with trivial symmetry group.

Dirk Frettl¨oh Symmetries of monocoronal tilings Open Problems

Regarding the third problem: there are non-periodic tricoronal tilings.

Dirk Frettl¨oh Symmetries of monocoronal tilings Dirk Frettl¨oh Symmetries of monocoronal tilings ∗ ∗ ∗

Thank you!

Open Problems

Dirk Frettl¨oh Symmetries of monocoronal tilings Open Problems

∗ ∗ ∗

Thank you!

Dirk Frettl¨oh Symmetries of monocoronal tilings